lecture 5 numbers and built in functions

1 of 45Module 5 : Numbers and Built‐in Functions

Introduction to Computational Thinking

Module 5 :

Numbers and Built‐in Functions

Asst Prof Chi‐Wing FU, PhilipOffice: N4‐02c‐104

email: cwfu[at]ntu.edu.sg


Topics• More on Numbers• Built-in functions• Comparing floating point numbers• Case Study: Finding Square Root• Self Study: Bitwise operators (optional)


More on Numbers• Integers and Floating point numbers

• Their representations in computers•Data range•Optional: Bitwise operators on integers


Memory: Bits and Bytes• 1 Byte = 8 bits (each bit either 0 or 1)• 1 Byte has 28 different variations, i.e., 256• How about 4 bytes?

0

0

0

0

0

0

0

0 0

0

00

1

11

1

1

1

1

1

1

1111000

1001

1002

Contents ofmemory

Memory

This is also covered in Introduction to Computing Systems


Integers (others)• In most programming languages, an

integer takes up 4 bytes. The first bit for sign: +ve or -ve, i.e., 31 bits left

• Since there is a zero needed to be represented (as all zeros in the 32 bits), the available data range is [-231, +231 – 1 ]


Integers (Python)• Integers in Python has unlimited precision• This is a useful feature in Python; otherwise,

we (programmers) have to handle them ourselves through programming effort


Float• For real numbers with decimals• Limited precision because we use limited

number of bits to store the data in a special format (you will learn later in other course):•Max. representable finite float•Min. positive normalized float

Want to know more? http://docs.python.org/tutorial/floatingpoint.htmlhttp://en.wikipedia.org/wiki/Double_precision_floating-point_format


Float• and it has limited precision…• Hence, computation results with floats

may not be exact


Integer VS Float• When writing programs, variables that are

numeric can be integers or floats• So… how to choose?

Think about the context!!!

Need decimal values? Want precision?


Integer VS Float• For the followings, which type is

more appropriate?•Age•Height•Volume of a container•Exchange rate•Voltage and current•Number of students in a class•Bank account balance


What are Built-in functions• Built-in functions are subroutines, procedures,

or methods that are built into Python itself; each provides a specific functionality

• See the Python standard library: http://docs.python.org/library/functions.html

• In fact, you knew some of them already:• print (for displaying data)• input (for reading user input)• float, int, str (for converting data)• type (for checking data type) • exec (for executing a string in Python), …, etc.


• abs(x) – Return absolute value of x

• min(x,y) and max(x,y) – Return minimum and maximum of x and y, respectively

They allow two or

more arguments

Let’s learn more: calling & arg(s)

Takes one argument


More Built-in Functions• math.ceil(x) and math.floor(x) –

Return the ceiling and floor of x as “int”

Remember to import math


More Built-in Functions• math.pow(x,y) – Return x raised to the power y• math.sqrt(x) – Return the square root of x• math.log(x) – Return natural logarithm of x• math.log10(x) – Return base-10 logarithm of x

Math functionsusually return“float”


More Built-in Functions• math.sin(x), math.cos(x), math.tan(x)

– Return the sine, cosine, and tangent of x (radian)

• math.asin(x),math.acos(x),math.atan(x)– Return arc sin, cos, and tan of x in radian(Note: they all use radian!)

• math.degrees(x) and math.radians(x) – Return x in degrees and radians, respectively


Help Function• help(“...”) – display help information on a

Python function or module


No Equality!• If we want to know if two floating point numbers

are equal or not, can we use "==" ??• It is tricky!!!

They are not equal! Why?So… how to fix it?


So… how to check?• First, we can define a relatively small floating

point value, usually called epsilon or tolerance• IDEA: if the absolute difference between the

two floating point numbers is smaller than epsilon, we say that their values are the same

Key idea!!!!

(computationally but not mathematically)


Case Study• Here we want to implement an algorithm to

compute the square root of a value without using math.sqrt

You should implement, run, and try the case study yourself in the course


Algorithm: compute square root• We can compute the square root of a value,

say x, iteratively using the following idea:Given x,

first compute an initial guess: guess = x/2then diff = guess* guess - xif abs(diff) < epsilon, we are doneif diff > 0, guess is too largeif diff < 0, guess is too smallupdate guess so that it gets closer, then

Repeat


Algorithm: compute square root• But… how to update guess so that it gets

closer in every iteration…• Let’s check x/guess…

In case x=100 and guess=50,x/guess = 2

the targeting square root value should always lie between guess and x/guess!

• So… we may update guess as mean value of guess and x/guess and make it closer!


Algorithm: compute square rootLet’s do this test…• Iteration #1

x=100 and guess=50x/guess=2 -> new guess = mean = 26

• Iteration #2x=100 and guess=26x/guess=3.846 -> new guess = mean = 14.923

• Iteration #3x=100 and guess=14.923x/guess=6.701 -> new guess = mean = 10.812…………

Really getting closer and closer!!!


Implementation• Here is the Python implementation

We will learn “while” in module 6.2


Implementation• Add variable "iter" to count number of iterations

Add iteration counter


Verification… by math.sqrt• Lastly… we must verify our program!!!

Report the differenceThis is a verification


Testing #1• We try with x = 100

Our result

Difference is very small


Testing #2• We try with x = 100000

Our result

But…need moreiterations forlarger num.

Differencestill small


Testing #3• Lastly… try with x = 2

Difference is still very small

Fewer iterationsthis time

Our result


What are Bitwise operators• Recall that integers are stored in binary, e.g.,


NOT~

SHIFT (right)>>

SHIFT (left)<<

Bitwise XOR^Bitwise OR|

Bitwise AND&meaningoperator

What are Bitwise operators• Python provides bitwise operators that process

input integers bit by bit correspondingly• Treat 1 as true and 0 as false

Input:15 -> 0000 111117 -> 0001 0001What is 15 | 17 ?

All are binaryoperations except ~,which is unary


00 & 0

00 & 1

01 & 0

11 & 1

ResultAND

Truth tables of &, |, ^, ~

00 | 0

10 | 1

11 | 0

11 | 1

ResultOR

00 ^ 0

10 ^ 1

11 ^ 0

01 ^ 1

ResultXOR

• XOR is called exclusive or; it is true ONLY if one of the two bits is true (but not both)

• ~x is a special operation called 2’s complement, which is just -x-1, i.e., ~2 gives -3


Examples: &, |, ^

&

|

^


Examples: shift <<, >>

<<

>>


3 << 1 gives 6SHIFT (left)<<

~3 gives -4NOT~

3 >> 1 gives 1SHIFT (right)>>

3 ^ 2 gives 1Bitwise XOR^

3 | 2 gives 3Bitwise OR|

3 & 2 gives 2Bitwise AND&examplesmeaningoperator

More examples

Try to work them out yourself


Any Application?1. Simple Data Encryption and Decryption

using the xor operator2. Pseudo Random Number Generator

(PRNG)


Application #1• xor can be used for doing simple encryption

and decryption• Given

D – our data (any integer, i.e., 32 bits)K – our key in encryption (which is an integer)

• We can perform encryption by:E = D xor K where E is the encrypted integer

• and decryption byE xor K gives D We can recover D from E using K

XOR has this interesting property Why? Study the truth table…


Application #2• Pseudo Random Number Generator (PRNG):

•Seed – A number to initialize the generator•Random number sequence – Use seed as

input, we apply the generator to obtain the first random number; then use it as input to generate the next random number, and so on…See http://docs.python.org/library/random.html

• In designing a PRNG, bitwise operations are often used, see next slide for a simple example

Interesting article:http://spectrum.ieee.org/semiconductors/processors/behind-intels-new-randomnumber-generator


Application #2 (cont.)• Example: Define a function “parityOf” to

compute the parity of an integer:A bit to add to an integer to make the number of 1s in its binary form to be even

Define a simple 12-bit PRNG

A sequence of random number(binary numbers) can begenerated starting from the seed

note: 0x indicateshexidecimal format


Application #2 (cont.)• Note:

• You will learn how to use Python keyword “def” to define functions later in this course

• Using bitwise operators, we can efficiently compute the parity and develop pseudorandom number generators. There are many PRNGs in the world; some are more complicated, and they could have different quality…

• Since the numbers generated in the example code form a binary number sequence, we also call it a Pseudo Random Binary Sequence Generator (PRBSG)

• Related courses in the future:“Computer Organization and Architecture” and “Microprocessor-based System Design”


Take Home Messages• Integers, Long, and Floating point numbers

– Representation and property: data range, minimum and maximum values, and precision

• Built-in functions are useful resources built into Python itself

• Comparing floating point numbers can be tricky (and error-prone)– Use absolute difference against a small epsilon to test

equality for floating point numbers


Reading Assignment • Textbook

Chapter 1: Beginnings1.8 to 1.10

Note: Though some material (1.10) in textbook is not directly related to the lecture material, you can learn more from them.

lecture 5 numbers and built in functions

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